Consider relationship between the two starting points for homographies and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.

Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q->1 ?

Relate Cayley's theorem to the field with one element.

Study:

How Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to Schur-Weyl duality.

There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs. There is none (negation=0).

Orthogonal group The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).... The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.

Clifford algebra Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

http://arxiv.org/pdf/0909.0069
[PL] J.L. Pena, O. Lorscheid: Mapping F1-land: An overview of geometries
over the field with one element, preprint, arXiv: mathAG/0909.0069.

New foundations for geometry Shai Haran.
This point of view of the general-linear-group suggests also that for the
field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which
embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u
n ¸ Sn). (the symmetry group for Bn and Cn)

Recently, there have been a few approaches to "geometry over F1", such as
Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s,
Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s.
For relations between these see rPLs.

F1-believers base their f-unny intuition on the following
two mantras :
• F1 forgets about additive data and retains only multiplicative data.
• F1-objects only acquire flesh when extended to Z (or C).

What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18],
deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.

Symmetry involves a dual point of view: for example, vertices are distinct and yet not distinguishable.

Of course, there is no field F1 with only one element, but there is a trivial ring,
and it is merely a convention that we do not call it a field. However, it is an excellent
convention, because the trivial ring has no nontrivial modules (if x is an element of
a module, then x = 1x = 0x = 0). Calling itj a field would not help solve Puzzle 1,
since F
n
1
does not depend on n.
I know of no direct solution to this puzzle, nor of any way to make sense of vector
spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes
much easier to understand when it is reformulated in terms of projective geometry.
That may not be surprising, if one keeps in mind that many topics, such as intersection
theory, become simpler when one moves to projective geometry. (The papers [11]
and [22] also shed light on this puzzle by indirect routes, but not by using projective
geometry.)

Cohn, page 489

Dear Harvey,

Thank you for your invitations in your letter below and also earlier, "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html

You mentioned, in my words, that you are looking for an issue that working mathematicians are grappling with where the classical ZFC foundations are not satisfactory or sufficient. Would the "field with one element" be such issue for you?
https://ncatlab.org/nlab/show/field+with+one+element

Jacques Tits raised this issue in 1957 and it has yet to be resolved despite substantial interest, conferences, and long papers. Would that count as a "problem" for the Foundations of Mathematics? It seems that in the history of math it is very easy to simply say "that is not real math" as was the case with the rational numbers, imaginary numbers, infinitesimals, infinite series, etc.

The issue is that there are many instances where a combinatorial interpretation makes sense in terms of a finite field Fq of characteristic q, which is all the more insightful when q=1. For example, the Gaussian binomial coefficients can be interpreted as counting the number of k-dimensional subspaces of an n-dimensional vector space over a finite field Fq. When q=1, then we get the usual binomial coefficients which count the subsets of size k of a set of size n. So this suggests an important way of thinking about sets. However, F1 would be a field with one element, which means that 0=1. But if 0 and 1 are not distinct, then none of the usual properties of a field make sense. Nobody has figured out a convincing interpretation for F1. And yet the concept seems to be pervasive, meaningful and fruitful.

If there was an alternate foundations of mathematics which yielded a helpful, meaningful, fruitful interpretation of F1, would that count in its favor? And if it could do everything that FOM can do, then might it be preferable, at least for some? But especially if that interpretation was shown not to make sense in other FOMs?

I am curious what you and others think.

Andrius

I just learned of the "field of one element":
https://en.wikipedia.org/wiki/Field_with_one_element
It's a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which apparently has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. I hope to learn more about it and report. But my impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?

Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”.
In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings. https://arxiv.org/pdf/1801.01491.pdf